Exercicios resolvidos James Stewart vol. 2 7ª ed - ingles

CHAPTER 15 PROBLEMS PLUS D 299
d
au au . A. • f) au . A. () h'l
An afJ = - ax psm.I'Slll + ay pSID'I'COS , W I e
Therefore
a 2 u a 2 u · a 2 u ·
-2 = -2 - a p 2 sin 2 if> cos() sin 0 + -a 2 p 2 sin 2 ¢ sin 2 ()
afJ y X X
a 2 u 2 au cot¢> au 1 8 2 u 1 a 2 u
- + --+ ---+ --+ ~--;:oap2
pap p2 a¢ p2 aq? p2 sin2 if> aB2
+ a2 u 2 . 2 A. 2 0 au . A. 0 au . A. • 0
a 2 p sm 'l'cos --a psm'l' cos - -a p sm 'l' sm
y X y
= ::~ [ (sin 2 ¢> cos 2 B) + ( cos 2 ¢> cos 2 B) + sin 2 B)
+ ~~ [ (sin 2 ¢> sin 2 0) + ( cos 2 ¢> sin 2 0) + cos 2 OJ + ~~ [cos 2 ¢> .+ sin 2 ¢)
+ au [ 2 sin 2 if> cos()+ cos 2 ¢> c~ 0 - sin 2 ¢> cos 0 - cos 0]
ax
psm¢>
+ au [ 2sin 2 ¢> sinB + cos 2 ¢> s~n()- sin 2 ¢> sin()- sinO]
ay
psm¢>
But 2 sin 2 ¢>cos 0 + cos 2 if> cos 0 - sin 2 ¢>cos 0 - cos 0 = (sin 2 if>+ cos 2 if> - 1) cos 0 = 0 and similarly the coefficient of
aujay is 0. Also sin 2 ¢co?> 2 0 + cos 2 if>cos 2 0 + sin 2 e = cos 2 fJ (sin 2 ¢ + cos 2 ¢) + sin 2 () = 1, and similarly the
coefficient of a 2 uj 8y 2 is 1. So Laplace's Equation in spherical coordinates is as stated.
11. Io"' I~ Io" f(t)dtdz dy = IIIE f(t)dV, where. y
· E = {(t, z, y) I 0 ~ t ~ z, 0 ~ z ~ y, 0 ~ y ~ x}. t =z
If we let. D be the projection of Eon the yt-plane then
D = {(y, t) I 0 S t S x, t S y S x}. And we see from the diagram
that E = { (t, z, y) I t ~ z ~ y, t ~ y ~ x, 0 ~ t ~ x }. So
} ~"' I~ I; f(t) dt dz dy = Io"' I/' .ftv f (t ) dz dy dt = Io"' [ft"'(y- t) f(t) dy] dt
= J-;' [(h 2 - ty)f(t) J~:; dt = Io"' [~x 2 - tx- ~t 2 + t 2 ]f(t) dt
= Io"' [~x 2 - tx + tt 2 ] f(t) dt = I o"' ax 2 - 2tx + t 2 )f(t) dt
= ~ I;(x - t) 2 f(t) dt
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